We focus here on a sub-family of parameterized algorithms, namely on kernelization. The goal of kernelization is to provide a polynomial algorithm transforming any instance \((I, k)\)  of the problem into an equivalent instance \((I^{\prime },k^{\prime })\)  where \(k^{\prime }\)  is upper bounded by some function of \(k\)  (in our case we will simply have \(k^{\prime } \leqslant k\) ), and the size of the new instance \(I^{\prime }\)  is upper bounded by some function \(g(k)\) . Hence the size of the reduced instance does not depend on the size of the original instance. Kernels are obtained throught a set of reduction rules.
While kernelization is possible for all FPT problems (the two notions are actually equivalent),
the interesting question is whether a given FPT problem admits polynomial kernels, where the size of the reduced instance is bounded by some polynomial in \(k\) . Note that, under some complexity assumptions, not all FPT problems admit polynomial kernels [1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
